Due to cold weather,a $1\, m$ water pipe of cross-sectional area $1\, cm^2$ is filled with ice at $-10^{\circ}C$. Resistive heating is used to melt the ice. $A$ current of $0.5\, A$ is passed through a $4\, k\Omega$ resistance. Assuming that all the heat produced is used for melting,what is the minimum time required? (In $s$)
(Given: latent heat of fusion for water/ice $= 3.33 \times 10^5\, J/kg$,specific heat of ice $= 2 \times 10^3\, J/(kg\cdot K)$ and density of ice $= 10^3\, kg/m^3$)

The two ends of a conductor are connected to a cell having an $e.m.f. \ E$ and some internal resistance. Starting from the midpoint $P$ of the conductor,moving in the direction of the current,and returning to point $P$,which of the following graphs best represents the potential $V$ versus the distance $x$ covered along the path?

The resistance of the filament of a lamp increases with the increase in temperature. $A$ lamp rated $100\, W, 220\, V$ is connected across a $220\, V$ power supply. If the voltage drops by $10\%$,then the power of the lamp will be:

Consider a block of conducting material of resistivity $\rho$ shown in the figure. Current $I$ enters at $A$ and leaves from $D$. We apply the superposition principle to find the voltage $\Delta V$ developed between $B$ and $C$. The calculation is done in the following steps: $(i)$ Take current $I$ entering from $A$ and assume it to spread over a hemispherical surface in the block. $(ii)$ Calculate the field $E(r)$ at distance $r$ from $A$ by using Ohm's law $E=\rho j$,where $j$ is the current per unit area at $r$. $(iii)$ From the $r$ dependence of $E(r)$,obtain the potential $V(r)$ at $r$. $(iv)$ Repeat $(i), (ii)$ and $(iii)$ for current $I$ leaving $D$ and superpose results for $A$ and $D$. $\Delta V$ measured between $B$ and $C$ is

For what value of the resistor $X$ will the equivalent resistance of the two circuits shown be the same?

Identify the correctness of the following statements:
$(1)$ The product of a volt and a coulomb is a joule.
$(2)$ The product of a volt and an ampere is a joule/second.
$(3)$ The product of a volt and a watt is a horsepower.
$(4)$ $A$ watt-hour can be measured in terms of an electron volt.